Meromorphic almost periodic functions

0.14 MB·English

by  S. Favorov

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February 2, 2008
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8 517.518.6 MSC 42A74 (30D35)
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Meromorphic almost periodic functions.

] N.Parfyonova, S.Ju.Favorov.
V
R
A function f C( ) is called almost periodic (a.p.) if for every ε > 0
C ∈
the set of ε-almost periods
.
h R R
t Eε(f)= τ : f(x+τ) f(x) <ε, x
a { ∈ | − | ∀ ∈ }
m R
is relatively dense in . The latter means that for some L > 0 and for
R
[ every a the segments [a,a+L] have common points with the set
∈
E (f). An analytic function f(z) on a horizontal strip S is said to be
1 ε
analytic a.p.function if for every ε > 0 and every substrip S′ S the set
v
2 of ε, S′-almost periods 1 ⊂⊂
2
R ′
8 Eε,S′(f)= τ : f(z+τ) f(z) <ε, z S
{ ∈ | − | ∀ ∈ }
1
R
0 is relatively dense in .
7 Theory of analytic a.p.functions was constructed by H.Bohr,
0
K.Bush, B.Jessen; for its detailed presentation, see [1, 2]. The fur-
/
h ther development of this theory is closely connected with the names of
t M.G.Krein, B.Ja.Levin, V.P.Potapov, H.Tornehave, L.I.Ronkin ([3]-
a
m [10]).
Let us give the following definition:
:
v A meromorphic function f(z) on a strip S (of finite or infinite width)
Xi is called meromorphic a.p.function on this strip if for each substrip S′ S
′ ⊂⊂
and each ε>0 the set of ε, S -almost periods
r
a
R ′
Eε,S′(f)= τ :ρ(f(z+τ),f(z))<ε, z S
{ ∈ ∀ ∈ }
R C
is relatively dense in ; by ρ we denote the spherical metric on =
C
.
∪{∞}
Notice that analytic a.p.functions are bounded on each substrip
′
S S and the spherical metric is equivalent to the Euclidean one
⊂⊂
on any bounded set; therefore any analytic a.p.function is a mero-
morphic a.p.function, too. It is easy to see that any uniformly con-
C
tinuous function on takes meromorphic a.p.functions to meromor-
phic a.p.functions. Hence each linear–fractional transformation takes a
1For S ={z ∈C: a<|Imz|<b}, −∞≤a<b≤∞, S′ ={z ∈C: α<|Imz|<β} the
encloseS′⊂⊂S meansthata<α<β<b.
1
meromorphic a.p.function (in particular, an analytic a.p.function) to a
meromorphic a.p.function.
We show that the class of meromorphic a.p.functions is not closed
under the operations of addition and multiplication. Nevertheless
the class of meromorphic a.p.functions inherits many properties of
the class of analytic a.p.functions: the uniform limit of meromorphic
a.p.functions is a meromorphic a.p.function; zeros and poles of mero-
morphic a.p.functions form almost periodic sets; Bochner’s criterion
of almost periodicity is valid for meromorphic a.p.functions. Also, we
show that every meromorphic a.p.function is a quotient of analytic
a.p.functions; this result is based on the theorems about a.p.discrete
sets from [11, 12]. Finally, using methods of [13, 14], we give a crite-
rion for a pair of a.p.sets to be the zero set and the pole set of some
meromorphic a.p.function.
At first we prove the following simple statement.
Theorem 1 A meromorphic a.p.function f(z) on a strip S is uniformly
′
continuous with respect to the metric ρ on any substrip S S.
⊂⊂
2 Take ε > 0. Let L be a positive number such that each segment
R ′
[a,a+L], a , contains an ε,S -almost period of f. A meromorphic
∈
function is uniformly continuous with respect to the spherical metric
′ ′
on every compact set, therefore for some δ > 0 and any z,z S z :
′ ′ ∈ ∩′ {
Rez < L+1 such that z z < δ we have ρ(f(z),f(z )) < ε. If z,z are
| | } ′ | − |′
arbitrary points in S and z z <δ, then there exists τ Eε,S′(f) such
| −′ | ∈
that Re(z+τ) <L+1, Re(z +τ) <L+1. We get
| | | |
′ ′ ′ ′
ρ(f(z),f(z )) ρ(f(z),f(z+τ))+ρ(f(z+τ),f(z +τ))+ρ(f(z +τ),f(z ))<3ε.
≤
Corollary 1 If f is a meromorphic a.p.function and N, P its zero set
and pole set respectively, then we have
′ ′ ′ ′ ′ ′
inf z z :z N S , z P S δ(S ,f)>0 S S. (1)
{| − | ∈ ∩ ∈ ∩ }≥ ∀ ⊂⊂
2 It follows from Theorem 1 that there are no sequences z , z′ S′ such
′ ′ n n ∈
that f(z )=0, f(z )= and z z 0 as n .
n n ∞ | n− n|→ →∞
We shall say that some property is valid inside S if it is valid on any
′
substrip S S. In particular, we shall say that sets N and P are
⊂⊂
separated inside S if they satisfy (1).
Now we can give examples of meromorphic a.p.functions such that
their sum or product is not a meromorphic a.p.function.
Take f (z) = sin√2πz, f (z) = 1/sinπz. By Kronecker’s Theorem (see
1 2
R Z
for example [2]), for any δ > 0 there exist t and m, n such that
t n < δ, t/√2 m < δ. The points z′ =∈ n/√2 and z′∈′ = m belong
| − | | − |
to the set of zeros and the set of poles, respectively, for the product
f (z)f (z) = sin√2πz/sinπz. Furthermore, we have n/√2 m n/√2
1 2
| − | ≤ | −
2
t/√2 + t/√2 m (1+1/√2)δ. Sincethechoiceofδ isarbitrary, zerosand
| | − |≤
poles of f (z)f (z) are not separated and f (z)f (z) is not a meromorphic
1 2 1 2
a.p.function. By the same argument, the sum 1/sinπz+1/sin(2√2 1)πz
−
is not a meromorphic a.p.function, too.
Howevertheclassofmeromorphica.p.functionsisclosedwithrespect
to the uniform convergence:
Theorem 2 If a sequence of meromorphic a.p.functions f (z) on S con-
n
verges uniformly inside S, then the limit function f(z) is a meromorphic
a.p.function on S.
2 It can easily be checked that the uniform limit of meromorphic func-
tionswithrespecttothesphericalmetricisalsoameromorphicfunction.
Now let n be large enough and τ Eε,S′(fn). Then we have
∈
ρ(f(z+τ),f(z)) ρ(f(z+τ),f (z+τ))+ρ(f (z+τ),f (z))+ρ(f (z),f(z))<3ε.
n n n n
≤
Hence f(z) is a meromorphic a.p.function on S.
The following result is very useful for the sequel.
Theorem 3 (Bochner’s criterion) The following conditions are equiva-
lent:
(i) f(z) is a meromorphic a.p.function on S;
′
(ii) for any sequence of real numbers t there exists a subsequence t
n n
′
such that the sequence of functions f(z+t ) converges uniformly inside
n
S.
2 The proof of this theorem is the same as the proof of Bochner’s crite-
rion for a.p.functions on the axis (see, for example, [2]).
Note that from Theorem 2 it follows that the sequence of functions
′
f(z+t ) converges to a meromorphic a.p.function inside S.
n
Let us prove the following statement.
Theorem 4 Let f(z) be a meromorphic a.p.function on S and let P be
r
the union of the disks of radius r with the centers at the poles of f(z);
′ ′
then f(z) is bounded on the set S P for any substrip S S.
r
\ ⊂⊂
2 Assume the contrary. Then there exists a sequence of points z =
n
′
x +iy S P such that f(z ) . Taking into account Theorem 3, we
n n r n
∈ \ →∞
may assume that the sequence of functions f(z+x ) converges uniformly
n
with respect to the spherical metric to a meromorphic a.p.function g(z)
inside S and the points iy converge to iy S. By uniform continuity
n 0
∈
of f(z) we get that the functions f(z+z iy ) also converge uniformly to
n 0
−
g(z). In particular, the point iy is a pole of g(z). Suppose B is a disk of
0 0
′
radius r <r withthe center iy such that B S and g(z)has no zeros on
0 0
⊂
B and no poles on ∂B ; then g(z) α>0 for z B . Hence the uniform
0 0 0
| |≥ ∈
convergence of the functions f(z +z iy ) to g(z) with respect to the
n 0
−
spherical metric implies the convergence of the functions 1/f(z+z iy )
n 0
−
3
to 1/g(z)with respect to the Euclidean metric. Hurwitz’ Theorem yields
that the functions f(z+z iy ) have poles w in B for n large enough.
n 0 n 0
−
So the points w +z iy are poles of f(z) and w +z iy z < r.
n n 0 n n 0 n
− | − − |
This contradicts the choice of z .
n
Let N be the union of the disks of radius r with the centers at the
r
zerosoff. ApplyingTheorem4tothefunction1/f,wegettheinequality
′
inf f(z) :z S N >0.
r
{| | ∈ \ }
Corollary 2 If a meromorphic a.p.function has no poles on S, then it
is an analytic a.p.function on S.
2 By the previous theorem, the function f(z) is bounded on each sub-
′ | |
strip S S. At the same time the spherical metric is equivalent to the
⊂⊂
Euclidean one on any bounded set.
Now we can give a simple condition for the product of meromorphic
a.p.functions to be a meromorphic a.p.function.
Theorem 5 Let f (z), f (z) be meromorphic a.p.functions. A necessary
1 2
and sufficient conditions for the product f (z)f (z) to be a meromorphic
1 2
a.p.function is that zeros and poles of this product be separated inside
S.
2 The necessity follows from Corollary 1. Let us prove the sufficiency.
Taking into account Theorem 3, we shall show that the uniform con-
vergence of f (z + t ) to g (z) and f (z + t ) to g (z) (with respect to
1 n 1 2 n 2
the spherical metric) inside S implies the uniform convergence of the
functions f (z+t )f (z+t ) to the function g (z)g (z).
1 n 2 n 1 2
First we shall show that the distance between zeros and poles of the
′
function g (z)g (z) in each substrip S S equals the distance between
1 2
⊂⊂
zeros and poles of the function f (z)f (z).
1 2
′ ′ ′′ ′′ ′ ′′ ′
Suppose g (z )g (z ) = 0, g (z )g (z ) = , z , z S S. Let
1 2 1 2
′ ′′ ∞ ∈ ⊂⊂ ′ ′′
C(z ), C(z ) be the circles of radius δ with the centers at the points z , z
respectively such that no zeros and poles of the functions g (z), g (z)
1 2
lie either on these circles or inside these circles, except for the centers.
′ ′ ′′ ′
It can be assumed also that C(z ) S , C(z ) S . Using Hurwitz’
⊂ ⊂
Theorem for the functions f (z+t ) and 1/f (z+t ) we obtain that for n
1 n 1 n
large enough, each function f (z+t ) has just the same number of zeros
1 n
′ ′′
and poles (with the multiplicity) inside the circles C(z ), C(z ) as the
′ ′′
function g (z) has at the points z , z respectively. The same assertion
1
is true for the functions f (z+t ) and g (z). Hence the number of zeros
2 n 2
′
of the product f (z +t )f (z +t ) inside the circle C(z ) is greater than
1 n 2 n
the number of its poles. Conversely, the number of poles of this product
′′
inside the circle C(z ) is greater than the number of its zeros. It follows
that the function f (z + t )f (z + t ) has at least one zero inside the
1 n 2 n
′ ′′
circle C(z ) and at least one pole inside the circle C(z ). Therefore the
distance between zeros and poles of the product f (z)f (z) is not greater
1 2
′ ′′
than z z +2δ with an arbitrary small δ.
| − |
4
On the other hand, the uniform convergence of g (z t ) to f (z) and
1 n 1
−
g (z t ) to f (z) inside S implies that the distance between zeros and
2 n 2
− ′ ′′
poles of the product g (z)g (z) is not greater than w w +2δ for an
1 2
′ ′ ′′ ′ | − |
arbitrary zero w S and a pole w S of the product f (z)f (z).
1 2
∈ ∈
Furthermore, supposeourtheoremisnottrue. Thenthereexistγ >0
′
and sequence of points z =x +iy S S such that
n n n
∈ ⊂⊂
ρ(f (z +t )f (z +t ),g (z )g (z )) γ. (2)
1 n n 2 n n 1 n 2 n
≥
Note that g (z), g (z) are meromorphic a.p.functions. Hence we can
1 2
assume without loss of generality that the functions g (z+x ) converge
j n
uniformly inside S to meromorphic a.p.functions h (z), j = 1,2, respec-
j
tively and the points iy converge to a point iy S. It is clear that
n 0
∈
the functions f (z +t +x ) converge uniformly as n to the same
j n n
→ ∞
functions h (z), 1,2. Since the functions f (z) are uniformly continuious,
j j
we see that f (z+z +t iy ) also converge uniformly inside S to the
j n n 0
−
functions h (z), j =1,2.
j
If the both functions h (z) are finite at the point iy , then the
j 0
sequences of functions f (z + z + t iy ) and g (z + z iy ), j =
j n n 0 j n 0
− −
1,2, are bounded at this point. Therefore the sequences of numbers
f (z +t )f (z +t ) and g (z )g (z ) have the common limit h (iy )h (iy ).
1 n n 2 n n 1 n 2 n 1 0 2 0
This contradicts inequality (2).
If the bothfunctions h (z)are non vanishingat the pointiy , thenthe
j 0
sequences of functions 1/f (z+z +t iy ) and 1/g (z+z iy ), j =1,2,
j n n 0 j n 0
− −
are boundedatthispoint. Therefore the sequencesofnumbers 1/[f (z +
1 n
t )f (z +t )] and 1/[g (z )g (z )] have the common limit 1/[h (iy )h (iy )].
n 2 n n 1 n 2 n 1 0 2 0
This also contradicts inequality (2).
Nowsupposethat the functionh (z)has azero ofmultiplicityk atthe
1
pointiy andthefunctionh (z)hasapoleofmultiplicityp katthesame
0 2
≤
point. LetC beacircle z : z iy =δ suchthatC S andthefunctions
0
{ | − | } ⊂
h , h have neither zeros nor poles on the set z : 0 < z iy δ .
1 2 0
{ | − | ≤ }
Hurwitz’ Theorem yields that each function f (z +z +t iy ), g (z +
1 n n 0 1
−
z iy ) has just k zeros inside C for n>n and so has g (z+z iy ). By
n 0 1 1 n 0
− −
the same reason, the functions f (z+z +t iy ), g (z+z iy ) have
2 n n 0 2 n 0
− −
just p poles inside C for n > n . Since zeros and poles of the functions
2
f (z)f (z) and g (z)g (z) are separated inside S, we see that only zeros of
1 2 1 2
the products f (z +x +t )f (z +x +t ), g (z +x )g (z +x ) can be in
1 n n 2 n n 1 n 2 n
the disk z : z iy <δ for δ small enough.
0
{ | − | }
Further, themoduliofthefunctionsf (z+x +t ), f (z+x +t ), g (z+
1 n n 2 n n 1
x ), g (z +x ) are bounded from above and bounded away from zero
n 2 n
uniformly on the circle C for n large enough. Hence for all z C and
∈
n>n ,
3
f (z+z +t iy )f (z+z +t iy ) g (z+z iy )g (z+z iy ) >γ.
1 n n 0 2 n n 0 1 n 0 2 n 0
| − − − − − |
By the Maximum Principle, we obtain that the same assertion is valid
for z =iy . This also contradicts inequality (2).
0
5
The same argument for the functions 1/[f (z + t )f (z + t )] and
1 n n 2 n n
1/[g (z )g (z )] leads to the contradiction with (2) in the case k < p.
1 n 2 n
Corollary 3 The quotient of two analytic a.p.functions on S is a mero-
morphic a.p.function on S iff zeros and poles of this quotient are sep-
arated inside S.
For studying zeros and poles of meromorphic a.p.functions we use
the concept of divisor.
C
Adivisor d inthe domain G isa setofpairs (a ,k ) such that the
n n
⊂ { }
support of the divisor suppd = a is a discrete set in G without limit
n n
{ } Z
points in G and multiplicities k are numbers from 0 . The divisor d of
n f
\{ }
a meromorphic function f on G is the set (a ,k ) such that a are zeros or
n n n
{ }
poles of f(z) and k are multiplicities of zeros and poles, with k >0 for
n n
| |
zeros and k <0 for poles. A divisor (a ,k ) can be identified with the
n n n
{ }
chargeconcentratedonthesetofpointsa withmassesk ; thechargefor
n n
the divisor of a meromorphic function f is equal to (1/2π) log f , where
△ | |
the Laplace operator is considered in the sense of distributions. The
′ ′ ′
sum of divisors d = a ,k , d = a ,k is a divisor with the support
′ { n n} { n n}
in a a and the corresponding multiplicities; in particular, the
{ n} ∪ { n} ′ ′ ′
multiplicity at the point a = a suppd suppd is equal to k +k .
m n ∈ ∩ m n
It is clear that d = d +d . Further, let d = (a ,k ) be a divisor;
fg f g n n
{ }
define d = (a , k ) : (a ,k ) d , d+ = (a , k ) : (a ,k ) d, k > 0 ,
n n n n n n n n n
d− = (|a|, k{): (|a ,|k ) d, k ∈<0}. Note{that| d|=d++d−,∈d+d− =d+}.
n n n n n
{ | | ∈ } | |
A divisor d in a strip S is called almost periodic if for any smooth
(infinite differentiabled) function χ(z) with the support in S the sum
k χ(a +t), i.e. the convolution of the charge d with the function χ,
P n n
R
is an a.p.function of t (see [10]).
∈
A divisor d= (a ,k ) is positive if k >0 for all n; this divisor can be
n n n
{ }
identified with the sequence b of points a where each a appears
m n n
{ } { }
k times. The sequence b in S is said to be almost periodic if for any
n m
′
S S and ε>0 the set
⊂⊂
R N N
Eε,S′ = t : thereexistsabijection α:
{ ∈ →
′ ′
such that b S &b S b +t b <ε
m α(m) m α(m)
∈ ∈ ⇒| − | }
R
isrelativelydensein (see[5]and, inaspecialcase, [3]). This definition
isequivalenttothe definitionofalmostperiodicityforthecorresponding
positive divisor (see [11, 12]).
Itwasprovedin[5]thatthedivisord ofeveryanalytica.p.functionf
f
is almost periodic. We extend this result to meromorphic f.p.functions:
Theorem 6 Suppose f is a meromorphic a.p.function on a strip S; then
its divisor d , divisor of zeros d+, divisor of poles d−, and divisor d are
f f f | |
almost periodic.
2 We need the following simple generalization of Lemma 3.1 from [10]:
6
N
Lemma 1 Let g, f , n , be meromorphic a.p.functions on a domain
n
C ∈
G . If ρ(f (z), g(z)) 0 uniformly on compact subset of G, then the
n
⊂ →
functions log f considered as the distributions on G converge to log g ,
n
| | | |
and the charges d considered as the distributions on G converge to the
fn
charge d .
g
2Itsufficestocheck theconvergence ofthe functionslog f tolog g ona
n
′ ′ | | | |
neighborhood of each point z G. If g(z )= , then g(z) is bounded on
′ ∈ 6 ∞
someneighborhoodU ofz . Hencethefunctionsf (z)convergeuniformly
n
on U to g(z) with respect to the Euclidean metric. Using lemma 3.1
from [10] we get the convergence of the distributions log f to log g on
n
′ | | | |
U. If g(z ) = , then the functions 1/f (z) converge uniformly on some
n
∞ ′
neighborhood of z to 1/g(z) with respect to the Euclidean metric and
we can use lemma 3.1 from [10] again. Note that d = (1/2π) log f
fn △ | n|
and d = (1/2π) log g ; since the differentiation keeps the convergence
g
△ | |
in the sense of distributions, we obtain the last assertion of the lemma.
We continue the proof of the theorem. Let us show that the convo-
lution
(df χ)(t)=Xknχ(an+t)
∗
n
of the charge d = a , k with an infinite differentiated function χ(z)
f n n
{ } R
with the support in S is an a.p.function of t .
∈
Let s beasequenceofrealnumbers. Taking into account Theorem
n
{ }
3, we may assume that the functions f(s +z) converge, with respect to
n
the spherical metric uniformly inside S, to a meromorphic a.p.function
g(z). Let us check that the functions (d χ)(s +t) converge uniformly
f n
R ∗
on t to the function (d χ)(t).
g
∈ ∗ R
Assume the contrary. Then for any δ >0 there exists a sequence tn′
such that
(df χ)(sn′ +tn′) (dg χ)(tn′) δ. (3)
| ∗ − ∗ |≥
As before, it can be assumed that the functions f(sn′+tn′+z) converge,
with respect to the spherical metric uniformly inside S, to a meromor-
phica.p.function h(z)and sodo the functions g(tn′+z). By the lemma, it
follows that the divisors of the functions f(sn′+tn′+z) converge in sense
of distributions to the divisor d and so do the divisors of the functions
h
g(tn′+z). Therefore the functions of t (df χ)(sn′+tn′+t), (dg χ)(tn′+t)
∗ ∗
converge to the same function (d χ)(t). This contradicts (3).
h
∗
Now it follows from Bochner’s criterion for a.p.functions on the axis
(see [2]) that (d χ)(t) is an a.p.function. Since this statement is true
f
∗
for all smooth and supported in S functions χ(z), we see that d is an
f
a.p.divisor.
Further, let φ(z) be a nonnegative smooth function with the support
′
inadiskB S S ofradiusr. Sincezerosandpolesoff areseparated
r
⊂ ⊂⊂ ′ R
inside S, we see that for r<r(S ) and for every t the support of the
∈
function φ(z+t) doesnot contains simultaneouslyzeros and polesoff(z).
7
Hence we have (d+ φ)(t)=max (d φ)(t),0 . Consequently the function
f ∗ { f∗ }
(d+ φ)(t) isana.p.functionoft. Since any smoothfunctionwithsupport
f ∗′
in S S is a linear combination of smooth functions with supports in
disks⊂o⊂f radius r < r(S′), we see that d+ is an a.p.divisor. Evidently,
f
d− =d +d+ and d =d +2d+ are also a.p.divisors.
f f f | f| f f
Corollary 4 Numbers of zeros and poles of meromorphic a.p.function
inside the rectangle
′ ′ R ′
Π (S ,t)= z S : Rez t <1 , t , S S,
1
{ ∈ | − | } ∈ ⊂⊂
′
are uniformly bounded from above by a constant depending on S only.
2 We shall prove that the numbers d(Π (S′,t)) are bounded uniformly
1
R | |
with respect to t . Let φ(z) be a nonnegative such that χ(z) = 1
′ ∈
on the set Π (S ,0)). Since the convolution (χ d)(t) is an a.p.function
1
R ∗| |
on , we see that this convolution is bounded. Therefore the numbers
′ R
d(Π (S ,t)) (χ d)(t) are bounded uniformly with respect to t .
1
| | ≤ ∗| | ∈
The following theorem with Corollary 3 gives the complete descrip-
tion of meromorphic a.p.functions.
Theorem 7 Any meromorphic a.p.function f(z) on a strip S is a quo-
tient of two analytic a.p.functions on S.
2 By Theorem 6, poles of f form the a.p.divisor d−. It follows from
f
′ − ′
[11, 12] that there exists an a.p.divisor d in S such that d +d is the
divisor of some a.p.analytic function h(z) on S. Then the function g(z)=
h(z)f(z) is a meromorphic function without poles. By Theorem 5 and
Corollary 2, we obtain that g(z) is an analytic a.p.function. So we have
f =g/h.
Corollary 5 Let f (z), f (z) be meromorphic a.p.functions. A necessary
1 2
and sufficient conditions for the sum f (z)+f (z) to be a meromorphic
1 2
a.p.function is that zeros and poles of this sum be separated inside S.
2 It follows from Theorem 7 that the sum f (z)+f (z) is a quotient of
1 2
two analytic a.p.functions; so the assertion follows from Corollary 3.
Consider the problem of realizability of an a.p.divisor as the divisor
of a meromorphic a.p.function. This problem was solved in [13, 14] for
positive a.p.divisors and analytic a.p.functions by the methods of coho-
mology theory. Namely, it was proved that to each positive a.p.divisor
d in a strip S a class of Cˇech cohomology c(d) of the group H2(KR,Z)
R
is assigned, KR being the universal Bohr compactification of ; c(d)=0
iff d is the divisor of an a.p.function on S, and c(d +d ) = c(d )+c(d ).
1 2 1 2
Moreover, the element c(d) remains the same for the restriction of d to
′
any S S and for the image of d under every homeomorphism of S onto
S˜ of th⊂e form
Γ(x+iy)=x+iγ(y). (4)
8
We do not give the definition of the Bohr compactification here (one
can find it in [15]). We need only that the group H2(KR,Z) can be
R R R R R
identified with the factor group Z = Z / a a : a (see
∧ ⊗ R { R⊗ ∈ }
[16]). For example, the element c(d) = λ µ Z corresponds to
the a.p.divisor dλµ with the support (λ+i∧µ)−1∈n +∧i(λ+iµ)−1n : n ,n
1 2 1 2
Z { ∈
with the multiplicities k = 1. If λ/µ is rational, then c(d) = 0;
} n1,n2
otherwise, c(dλµ) = 0 and the divisor dλµ is the divisor of no analytic
6
a.p.function; this coincides with the corresponding result of [6].
R R
Note that every element of Z , in particular c(d), is a finite sum
∧
of elements λ µ with irrational λ/µ.
∧
Theorem 8 A divisor d in a strip S is the divisor of a meromorphic
a.p.function on S if and only if the following conditions are fulfilled:
i) the supports of the divisors d+ and d− are separated inside S,
ii) d+,d− are a.p.divisors,
iii) c(d+)=c(d−).
2 If d=d for some meromorphica.p.function on S, then i) follows from
f
Corollary 1, ii)follows from Theorem 6. Now by Theorem 7 we have f =
g/h, where g, h are analytic a.p.function on S, therefore c(d )=c(d )=0.
g h
Since d =d++d′, d =d−+d′ for any a.p.divisor d′ in S, we obtain iii).
g f h f
n
Ontheotherhand,supposei),ii),iii)arefulfilled;thenc(d+)= λ
P j
∧
j=1
µj with irrational λj/µj, 1 ≤ j ≤ n. By d′ denote the sum Pnj=1dµjλj;
n
then c(d+ +d′) = c(d+)+ µ λ = 0, hence there exists an analytic
P j j
∧
j=1
a.p.function g(z) on S such that d =d++d′; By iii) we have c(d−+d′)=
g f
c(d+ +d′) = 0, therefore d− +d′ = d for some analytic a.p.function h(z)
h
on S. It follows from i) that f = g/h is a meromorphic a.p.function on
S. Finally, d =d d =d.
f g h
−
Corollary 6 Suppose d is a divisor in S such that conditions i) and ii)
′
of Theorem 8 are fulfilled. If the restriction of d to S S is the divisor
′ ⊂
of some meromorphic a.p.function on S , then d is the divisor of a
meromorphic a.p.function on S; if Γ is a homeomorphism of S onto S˜
of the form (4) and d is the divisor of some meromorphic a.p.function
on S, then Γd is the divisor of some meromorphic a.p.function on S˜.
9
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